There may be a fancy way but its easier to do something like this:
You already have the global turning points from the derivative you mention. if there is no turning point within the range [0,1] or [0,2], then the maximal point must be at one of the boundaries. Compute the value at each boundary and pick the largest.
If there are turning points within the range then the maximum is either at the turning point or the boundaries, follow the same proceedure.